Radio‐tracking gravel particles in a large braided river in New Zealand: a field test of the stochastic theory of bed load transport proposed by Einstein

Hans A. Einstein initiated a probabilistic approach to modelling sediment transport in rivers. His formulae were based on theory and were stimulated by laboratory investigations. The theory assumes that bed load movement occurs in individual steps of rolling, sliding or saltation and rest periods. So far very few attempts have been made to measure stochastic elements in nature. For the first time this paper presents results of radio‐tracing the travel path of individual particles in a large braided gravel bed river: the Waimakariri River of New Zealand. As proposed by Einstein, it was found that rest periods can be modelled by an exponential distribution, but particle step lengths are better represented by a gamma distribution. Einstein assumed an average travel distance of 100 grain‐diameters for any bed load particle between consecutive points of deposition, but larger values of 6·7 m or 150 grain‐diameters and 6·1 m or 120 grain‐diameters were measured for two test particle sizes. Together with other available large scale field data, a dependence of the mean step length on particle diameter relative to the D₅₀ of the bed surface was found. During small floods the time used for movement represents only 2·7% of the total time from erosion to deposition. The increase in percentage of time being used for transport means that it then has to be regarded in stochastic transport models. Tracing the flow path of bed load particles between erosion and deposition sites is a step towards explaining the interactions between sediment transport and river morphology. Copyright © 2001 John Wiley & Sons, Ltd.     

Influence of bed clusters and size gradation on operational time distribution for non‐uniform bed‐load transport

The operational time distribution (OTD) defines the time for bed‐load sediment spent in motion, which is needed to characterize the random nature of sediment transport. This study explores the influence of bed clusters and size gradation on OTD for non‐uniform bed‐loads. First, both static and mobile bed armouring experiments were conducted in laboratorial flumes to monitor the transport of mixed sand/gravel sediments. Only in the mobile armouring experiment did apparent bed clusters develop, because of stable feeding and a longer transport period. Second, a generalized subordinated advection (GSA) model was applied to quantify the observed dynamics of tracer particles. Results show that for the static armour layer (without sediment feed), the best‐fit OTD assigns more weight to the large displacement of small particles, likely because of the size‐selective entrainment process. The capacity coefficient in the GSA model, which affects the width of the OTD, is space dependent only for small particles whose dynamics can be significantly affected by larger particles and whose distribution is more likely to be space dependent in a mixed sand and gravel system. However, the OTD for the mobile armour layer (with sediment recirculation) exhibited longer tails for larger particles. This is because the trailing edge of larger particles is more resistant to erosion, and their leading front may not be easily trapped by self‐organized bed clusters. The strong interaction between particle–bed may cause the capacity coefficient to be space‐dependent for bed‐load transport along mobile armour layers. Therefore, the combined laboratory experiments and stochastic model analysis show that the OTD may be affected more by particle–bed interactions (such as clusters) than by particle–particle interactions (e.g. hiding and exposing), and that the GSA model can quantify mixed‐size sand/gravel transport along river beds within either static or mobile armour layers. Copyright © 2016 John Wiley & Sons, Ltd.     

Linkages between bedload displacements and topographic change

Changes in bed topography that build and maintain channel morphology are driven by the displacements of individual particles, either though their entrainment or deposition. However, the linkages between these topographic changes and individual grain displacements have not been comprehensively addressed, as many historical tracer studies have not included coincident topographic data. In this study, we compare the movements of bedload tracers to the differences in repeat topographic surveys across four gravel-bed river reaches. To do this, we apply a 1-D Bayesian survival process model to the starting and ending locations of tracers. This model estimates downstream trapping probabilities, which represent the likelihood that a given segment of channel will “trap” an entrained particle. We then adapt this model to estimate downstream trapping probabilities using digital elevation models of difference and compare the results. The estimates from the tracer and topographic trapping models showed general alignment, meaning that tracers were preferentially trapped in segments that experienced deposition along the channel. Thus, tracers in this study were able to identify downstream differences in bedload transport. The comparison also highlighted that tracer-estimated trapping probabilities were larger than topographically estimated ones. This supports previous observations that sediment travel distances estimated using tracers are shorter than those estimated using morphological methods. We find that the differences between these two estimates vary systematically across study environments. These variations are attributable to either study design (i.e., tracers being larger than the median size of the sediment that deforms the bed) or differences in compensating scour and fill. We explore potential causes for differences in compensating scour and fill, including hydrograph shape, sediment delivery regime, channel deformation style, and channel width, highlighting that morphodynamics needs to be considered in designing bedload tracer studies.     

The role of channel morphology on the mobility and dispersion of bed sediment in a small gravel‐bed stream

This study uses a unique 10‐year tracer dataset from a small gravel‐bed stream to examine bed mobility and sediment dispersion over long timescales and at a range of spatial scales. Seasonal tracer data that captured multiple mobilizing events was examined, while the effects of morphology on bed mobility and sediment dispersion were captured at three spatial scales: within morphological units (unit scale), between morphological units (reach scale) and between reaches with different channel morphologies (channel scale). This was achieved by analyzing both reach‐average mobility and travel distance data, as well as the development of ‘mobility maps’ that capture the spatial variability in tracer mobility within the channel. The tracer data suggest that sediment transport in East Creek remains near critical the majority of the time, with only rare large events resulting in high mobility rates and grain travel distances large enough to move sediment past dominant bedforms. While a variable capturing both the magnitude and frequency of flow events within a season yielded a better predictor to sediment mobility and dispersion than peak discharge alone, the distribution of events of different magnitude within the season played a large role in determining tracer mobility rates and travel distances. The effects of morphology differed depending on the analysis scale, demonstrating the importance of scale, and therefore study design, when examining the effect of morphology on sediment transport. Copyright © 2016 John Wiley & Sons, Ltd.     

Particle travel distances and bed and sediment compositions associated with rain‐impacted flows

Laboratory experiments were performed with rain of uniform drop size (2·7 mm, 5·1 mm) impacting flows over non‐cohesive beds of uniform sized sand (0·11–0·9 mm) and coal (0·2–0·9 mm) particles with flow velocities (20 mm s^?1, 40 mm s^?1) that were insufficient for the flow to entrain the particles without the aid of raindrop impact. Measurement of particle travel distance under rain made up of 2·7 mm drops confirmed a theoretical relationship between settling velocity and the distance particles travel after being disturbed by drop impact. Although, in theory, a relationship between settling velocity and particle travel distance exists, settling velocity by itself was unable to account for the effect of changes in both particle size and density on sediment discharge from beds of uniform non‐cohesive material. Particle density was also a factor. Further study of how particle characteristics influence sediment discharge will aid modelling of the impact of the soil in process‐based models of erosion by rain‐impacted flow. Copyright © 2001 John Wiley & Sons, Ltd.     

A Bed-load function based on kinetic theory

Bed-load transport plays a fundamental role in morphological processes of natural rivers and human-made channels.This paper presents bed-load function derived on the basis of kinetic theory. The bed-load function is obtained by integrating the pick-up rate of bed sediment with respect to its longitudinal travel distance,following the basic concept on bed-load put forward by Einstein.The pick-up rate is expressed as an upwards flux of bed sediment and determined by invoking the particle velocity distribution function derived by solving the Boltzmann equation of kinetic theory. Comparisons of the present formula with six other bed-load formulas and the experimental data are also made in this paper.The results show that the present bed-load formula agrees well with the experimental data.In addition,the influences due to collision between particles on bed-load is discussed which shows that an appreciable damping of the intensities of bed-load is observed only when the shear acting on particles is large enough to increase the concentration and intensity of random motion of bed load particles so that the collisions between sediment particles can occur.     

Scale relationships in hillslope runoff and erosion

Eight runoff plots, located within a small catchment within the Walnut Gulch Experimental Watershed, southern Arizona, were constructed to test the argument that sediment yield (kg m^?2) decreases as plot length increases. The plots ranged in length from 2 m to 27·78 m. Runoff and sediment loss from these plots were obtained for ten natural storm events. The pattern of sediment yield from these plots conforms to the case in which sediment yield first increases as plot length increases, but then subsequently decreases. Data from the present experiment indicate that maximum sediment yield would occur from a plot 7 m long. Analysis of both runoff and sediment yield from the plots indicates that the relationship of sediment yield to plot length derives both from the limited travel distance of individual entrained particles and from a decline in runoff coefficient as plot length increases. Particle‐size analysis of eroded sediment confirms the role of travel distance in controlling sediment yield. Whether in response to the finite travel distance of entrained particles or the relationship of runoff coefficient to plot length, the experiment clearly demonstrates that the erosion rates for hillslopes and catchments cannot be simply extrapolated from plot measurements, and that alternative methods for estimating large‐area erosion rates are required. Copyright © 2006 John Wiley & Sons, Ltd.     

Fluid–particle interaction in turbulent open channel flow with fully-resolved mobile beds

The paper presents Direct Numerical Simulations of an open channel flow laden with spherical particles at a bulk Reynolds number of 2941. The transport of thousands of mobile particles is simulated propagating over a rough bed which consists of immobile particles of the same size in hexagonal ordering. An Immersed Boundary Method is used for the numerical representation of the particles. With 22 points per diameter even the viscous scales of the flow are resolved at this Reynolds number. The reference run contains just as many fixed as mobile particles with a relative density slightly above the nominal threshold of incipient motion. Further runs were conducted with decreased mass loading and decreased Shields number together with a simulation containing only immobile particles. The variation of the parameters defining the mobile sediment yields a strong modification of particle–fluid as well as particle–particle interactions yielding different structures in space and time. This is assessed by means of appropriate statistical quantities addressing the continuous and the disperse phase. The results are in qualitative agreement with experimental observations at higher Reynolds number.     

A flume experiment on wood storage and remobilization in braided river systems

This work investigates wood dynamics in braided streams through physical modelling in a mobile bed laboratory flume, with the specific objective to characterize wood storage and turnover as a function of wood input rate and of wood element type. Three parallel channels (1.7 m wide, 10 m long) filled with uniform sand were used to reproduce braided networks with constant water discharge and sediment feeding. Wood dowels with and without simplified root wads were regularly added at the upstream end of each flume at different input rates, with a 1:2:3 ratio between the three flumes. Temporal evolution of wood deposition patterns and remobilization rates were monitored by a series of vertical images that permitted the recognition of individual logs. Results show that wood tends to disperse in generally small accumulations (< 5 logs), with higher spatial density on top of sediment bars, and is frequently remobilized due to the intense morphological changes. The amount of wood stored in the channel depends on log input rate through a non‐linear relationship, and input rates exceeding approximately 100 logs/hour determine a sharp change in wood dynamics, with higher storage volume and augmented formation of large jams (> 10 elements) that are less prone to remobilization. Presence of root wads seems to play a minor role in wood deposition, but it reduces the average travel distance of logs. Turnover rates of logs were similar in the three flumes, independently of wood input rate and largely resembling the turnover rate of exposed bars. For the simulated conditions, significant effects of wood on bed morphology were not observed, suggesting that interactions with fine sediments and living vegetation are crucial to form large, stable wood jams able to bring about relevant morphological changes. Copyright © 2014 John Wiley & Sons, Ltd.     

3D Lagrangian modelling of saltating particles diffusion in turbulent water flow

A 3D Lagrangian model of the saltation of solid spherical particles on the bed of an open channel flow, accounting for turbulence-induced mechanisms, is proposed and employed as the key tool of the study. The differences between conventional 2D models and a proposed 3D saltation model are discussed and the advantages of the 3D model are highlighted. Particularly, the 3D model includes a special procedure allowing generation of 3D flow velocity fields. This procedure is based on the assumption that the spectra of streamwise, vertical and transverse velocity components are known at any distance from the bed. The 3D model was used to identify and quantify effects of turbulence on particle entrainment and saltation. The analysis of particle trajectories focused on their diffusive nature, clarifying: (i) the effect of particle mobility parameter; (ii) the effect of bed topography; and (iii) the effect of turbulence. Specifically, the results of numerical simulations describing the abovementioned effects on the change in time of the variance are presented. In addition, the change in time of the skewness and kurtosis, which are likely to reflect the turbulence influence on the spread of particles, are also shown. Two different diffusion regimes (local and intermediate) for each of the investigated flow conditions are confidently identified.     

The particle size characteristics of fine‐grained channel deposits in the River Exe Basin,Devon, UK

Deposition and storage of fine‐grained (<62·5 μm) sediment in the hyporheic zone of gravel bed rivers frequently represents an important cause of aquatic habitat degradation. The particle size characteristics of such fine‐grained bed sediment (FGBS) exert an important control on its hydrodynamic properties and environmental impact. Traditionally, particle size analysis of FGBS in gravel bed rivers has focused on the absolute size distribution of the chemically dispersed mineral fraction. However, recent work has indicated that in common with fluvial suspended sediment, significant differences may exist between the absolute and the in situ, or effective, particle size composition of FGBS, as a result of the existence of aggregates, or composite particles. In the investigation reported in this paper, sealable bed traps that could be remotely opened to sample sediment deposited during specific storm runoff events and a laser back‐scatter probe were used to quantify the temporal and spatial variability of both the absolute and effective particle size composition of FGBS, and the associated suspended sediment from four gravel bed rivers in the Exe Basin, Devon, UK. The absolute particle size distributions of both the FGBS and suspended sediment evidenced c. >95%<62·5 μm sized primary particles and displayed a seasonal winter–summer fining, while the opposite trend was displayed by the effective particle size distribution of the FGBS and suspended sediment. The effective particle size distributions of both were typically highly aggregated, comprising up to 68%>62·5 μm sized particles. Spatial variation in the effective particle size and aggregation parameters was of secondary importance relative to temporal variation. The effective particle size distribution of the FGBS was consistently coarser and more aggregated than the associated suspended sediment and there was evidence of aggregate break‐up in samples of resuspended bed sediment. The implications of these findings for sediment transport modelling are considered. Copyright © 1999 John Wiley & Sons, Ltd.     

Particle velocity and concentration profiles in bedload experiments on a steep slope

Depth profiles of particle streamwise velocity, concentration and bedload sediment transport rate were measured in a turbulent and supercritical water flow. One‐size 6 mm diameter spherical glass beads were transported at equilibrium in a two‐dimensional 10% steep channel with a mobile bed. Flows were filmed from the side by a high‐speed camera. Particle tracking algorithms made it possible to determine the position, velocity and trajectory of a very large number of particles. Approximately half of the sediment transport rate was composed by rolling grains, and the other half by saltation. This revealed a complex structure, with several concentration and flux peaks due to rolling, and one peak due to saltation. With an increase of the sediment transport rate, the depth structure remained the same at the water/granular interface, with peak value increases but with no shift in elevations. The saltation region expanded towards higher elevations with an increase of the particle velocity commensurate to the water velocity. The proportion of the sediment transport rate in saltation did not vary significantly. The particle streamwise velocity profiles exhibited three segments: an exponential decay in the bed, a linear increase where rolling and saltation co‐existed, and above this, a logarithmic‐like shape due to saltating particles. These results are comparable to profiles measured and modelled in dry granular free surface flows and in more intense bedload such as sheet flows. Copyright © 2014 John Wiley & Sons, Ltd.     

Virtual rate and mean distance of travel of individual clasts in gravel-bed channels

Travel distances in gravel-bed rivers of tagged particles of various sizes are related to excess stream power estimated for peak discharge. Mean distance of movement, irrespective of grain size, is weakly correlated with stream power, especially near the threshold of movement. There may be several reasons for the weak correlation, including variable effects of bed structure, varying magnitudes of sediment mobilizing events and sampling problems. Grain size itself is of marginal significance. The virtual rate of travel calculated using total time for which the flow is larger than that needed to initiate clast movement also bears a weak relation to the excess stream power over the period. Better results are obtained by relating the virtual rate of travel to the first peak of the flow event only. This implies that the initial seeding of the tagged particles dominates the observations. Nonetheless, an underlying general relation is indicated by the data, which are derived from a wide range of flow regime types.     

Assessment of erosion and settling properties of fine sediments stored in cobble bed rivers: the Arc and Isère alpine rivers before and after reservoir flushing

Cohesive sediment dynamics in mountainous rivers is poorly understood even though these rivers are the main providers of fine particles to the oceans. Complex interactions exist between the coarse matrix of cobble bed rivers and fine sediments. Given that fine sediment load in such environments can be very high due to intense natural rainfall or snowmelt events and to man‐induced reservoir or dam flushing, a better understanding of the deposition and sedimentation processes is needed in order to reduce ecohydrological downstream impacts. We tested a field‐based approach on the Arc and Isère alpine rivers combining measurements of erosion and settling properties of river bed deposits before and after a dam flushing, with the U‐GEMS (Gust Erosion Microcosm System) and SCAF (System Characterizing Aggregates and Flocs), respectively. These measurements highlight that critical shears, rates of erosion, settling velocities and propensity of particles to flocculate are highly variable in time and space. This is reflective of the heterogeneity of the hydrodynamic conditions during particle settling, local bed roughness, and nature and size of particles. Generally the deposits were found to be stable relative to what is measured in lowland rivers. It was, however, not possible to make a conclusive assessment of the extent to which the dynamics of deposits after reservoir flushing were different from those settled after natural events. The absence of any relationships between erosion and deposition variables, making it impossible to predict one from another, underlined the need to measure all of them to have a full assessment of the fine sediment dynamics and to obtain representative input variables for numerical models. While the SCAF was found to be effective, an alternative to the U‐GEMS device will have to be found for the erodibility assessment in cobble bed rivers, in order to make more rapid measurements at higher shears. Copyright © 2017 John Wiley & Sons, Ltd.     

Sediment connectivity: a framework for understanding sediment transfer at multiple scales

A major challenge for geomorphologists is to scale up small‐magnitude processes to produce landscape form, yet existing approaches have been found to be severely limited. New ways to scale erosion and transfer of sediment are thus needed. This paper evaluates the concept of sediment connectivity as a framework for understanding processes involved in sediment transfer across multiple scales. We propose that the concept of sediment connectivity can be used to explain the connected transfer of sediment from a source to a sink in a catchment, and movement of sediment between different zones within a catchment: over hillslopes, between hillslopes and channels, and within channels. Using fluvial systems as an example we explore four scenarios of sediment connectivity which represent end‐members of behaviour from fully linked to fully unlinked hydrological and sediment connectivity. Sediment‐travel distance – when combined with an entrainment parameter reflecting the frequency–magnitude response of the system – maps onto these end‐members, providing a coherent conceptual model for the upscaling of erosion predictions. This conceptual model could be readily expanded to other process domains to provide a more comprehensive underpinning of landscape‐evolution models. Thus, further research on the controls and dynamics of travel distances under different modes of transport is fundamental. Copyright © 2014 John Wiley & Sons, Ltd.     

Through‐water terrestrial laser scanning of gravel beds at the patch scale

Acquiring high resolution topographic data of natural gravel surfaces is technically demanding in locations where the bed is not exposed at low water stages. Often the most geomorphologically active surfaces are permanently submerged. Gravel beds are spatially variable and measurement of their detailed structure and particle sizes is essential for understanding the interaction of bed roughness with near‐bed flow hydraulics, sediment entrainment, transport and deposition processes, as well as providing insights into the ecological responses to these processes. This paper presents patch‐scale laboratory and field experiments to demonstrate that through‐water terrestrial laser scanning (TLS) has the potential to provide high resolution digital elevation models of submerged gravel beds with enough detail to depict individual grains and small‐scale forms. The resulting point cloud data requires correction for refraction before registration. Preliminary validation shows that patch‐scale TLS through 200 mm of water introduces a mean error of less than 5 mm under ideal conditions. Point precision is not adversely affected by the water column. The resulting DEMs can be embedded seamlessly within larger sub‐aerial reach‐scale surveys and can be acquired alongside flow measurements to examine the effects of three‐dimensional surface geometry on turbulent flow fields and their interaction with instream ecology dynamics. Copyright © 2011 John Wiley & Sons, Ltd.     

The influence of channel morphology on bedload path lengths: Insights from a survival process model

Tracer studies are a commonly used tool to develop and test Einstein-type stochastic bedload transport models. The movements of these tracers are controlled by many factors including grain characteristics, hydrologic forcing, and channel morphology. Although the influence of these sediment storage zones related to morphological features (e.g., bars, pools, riffles) have long been observed to “trap” bedload particles in transport, this influence has not been adequately quantified. In this paper we explore the influence of channel morphology on particle travel distances through the development of a Bayesian survival process model. This model simulates particle path length distributions using a location-specific “trapping probability” parameter (p_i ), which is estimated using the starting and ending locations of bedload tracers. We test this model using a field tracer study from Halfmoon Creek, Colorado. We find that (1) the model is able to adequately recreate the observed multi-modal path length distributions, (2) particles tend to accumulate in trapping zones, especially during large floods, and (3) particles entrained near a trapping zone will travel a shorter distance than one that is further away. Particle starting positions can affect path lengths by as much as a factor of two, which we confirm by modelling “starting-location-specific” path length probability distributions. This study highlights the importance of considering both tracer locations and channel topography in examinations of field tracer studies. © 2020 John Wiley & Sons, Ltd.